The conjugate descent (CD) method is a conjugate gradient (CG) variant with remarkable convergence properties. This paper presents a modified CD scheme for solving systems of constrained monotone nonlinear equations. Eigenvalue analysis demonstrates that the proposed search direction matrix is positive definite. By incorporating the proposed algorithm with Solodov and Svaiter's projection method (1999), several convex-constrained monotone nonlinear equations were solved, yielding impressive results. The new algorithm exhibits descent properties and achieves global convergence under appropriate assumptions. Numerical comparisons with recent algorithms in the literature highlight the efficiency and effectiveness of the proposed method. Furthermore, the method is applied to signal recovery experiments in compressive sensing.

This paper introduces a matrix-free variant of the Davidon-Fletcher-Powell (DFP) method for unconstrained optimization problems with applications in compressive sensing and image restoration. The main contribution lies in the new search direction incorporating a scaling parameter that ensures the satisfaction of the sufficient descent condition, independent of the line search conditions. A rigorous convergence analysis guarantees the boundedness and theoretical validity of the proposed method. Comprehensive numerical experiments on benchmark unconstrained optimization test problems and compressive sensing problems demonstrate the efficiency and robustness of the algorithm. Specifically, in image restoration tasks, our method outperforms CG-DESCENT, MDL, and NSMA, achieving a 100% success rate compared to 95.8%, 84.5%, and 53.5%, respectively. Additionally, results on computational time, relative error, and PSNR confirm the superior performance of the proposed approach. These findings establish the proposed method as a competitive alternative for large-scale optimization problems.